TITLE: Chromatic number of a graph defined by $n$ lines on the plane
QUESTION [10 upvotes]: Given $n$ lines on the plane, consider all their intersection points. Find the minimal number $d=d(n)$ such that they may be always colored in $d$ colors so that on each line any two consecutive points have different color. Of course, $d(n)\leqslant 4$ as this graph is planar. But is it true that $d(n)=3$ for all $n\geqslant 3$?
There is an old olympiad problem that 3 colors are enough if no three lines are concurrent. I gave it to children forgetting this condition, and so now I wonder whether is this still true.

REPLY [11 votes]: No, here is an example that needs four colours (if I have understood the question correctly):

(there are other intersection points not shown, of course, but these are irrelevant)